MinT - Best Known (86−61, 86, s)-Nets in Base 9

Best Known (86−61, 86, s)-Nets in Base 9

(86−61, 86, 78)-Net over F₉ — Constructive and digital

Digital (25, 86, 78)-net over F₉, using

t-expansion [i] based on digital (22, 86, 78)-net over F₉, using
- net from sequence [i] based on digital (22, 77)-sequence over F₉, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 22 and N(F) ≥ 78, using
    - F₃ from the tower of function fields by GarcÃa and Stichtenoth over F₉ [i]

(86−61, 86, 96)-Net over F₉ — Digital

Digital (25, 86, 96)-net over F₉, using

net from sequence [i] based on digital (25, 95)-sequence over F₉, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 25 and N(F) ≥ 96, using
  - a curve from the manYPoints database [i]

(86−61, 86, 742)-Net in Base 9 — Upper bound on s

There is no (25, 86, 743)-net in base 9, because

1 times m-reduction [i] would yield (25, 85, 743)-net in base 9, but
- the generalized Rao bound for nets shows that 9^mÂ â‰¥ 1296 168045 598257 917799 651284 940019 542601 624505 376355 518606 727499 712694 312574 912913 > 9⁸⁵ [i]